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On Chain Graph Models For Description Of Conditional Independence Structures
 Ann. Statist
, 1998
"... This paper deals with chain graphs (CGs) which allow both directed and undirected edges. This class of graphs, introduced by Lauritzen and Wermuth [15], generalizes both UGs and DAGs. To establish the semantics of CGs one should associate an independency model to every CG. Some steps were already ma ..."
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Cited by 19 (3 self)
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This paper deals with chain graphs (CGs) which allow both directed and undirected edges. This class of graphs, introduced by Lauritzen and Wermuth [15], generalizes both UGs and DAGs. To establish the semantics of CGs one should associate an independency model to every CG. Some steps were already made. Lauritzen and Wermuth [16] intended to use CGs to describe independency models for strictly positive probability distributions and introduced the concept of the chain Markov property which is analogous to the concept of causal input list for DAGs. Lauritzen and Frydenberg [17, 9] generalized the concept of moral graph and introduced a moralization criterion for reading independency statements from a CG. Frydenberg [9] characterized CGs with the same Markov ON CHAIN GRAPH MODELS 3 property (that is producing the same CGmodel) and Andersson, Madigan and Perlman [3] used special CGs to represent uniquely classes of Markov equivalent DAGs. Whittaker [31] in his book gave several examples of the use of CGs, and other recent works also deal with them [6, 20, 23, 30], the most comprehensive account is provided by the book [19]. Several results proved here were already presented (without proof) in our previous conference contribution [5]. An alternative approach to the generalization of UGs and DAGs was started by Cox and Wermuth [7] who introduced a wider class of jointresponse chain graphs which allow also 'dashed' directed and undirected edges in addition to the classic 'solid' directed and undirected edges treated in this paper. Andersson, Madigan and Perlman [1] introduced an alternative Markov property to give an interpretation to those jointresponse CGs which combine dashed directed edges with solid undirected edges (of course, another independency model is associated...
On Recovery Algorithm for Chain Graphs
, 1997
"... The class of chain graphs (CGs) involving both undirected graphs (= Markov networks) and directed acyclic graphs (= Bayesian networks) was introduced in middle eighties for description of probabilistic conditional independence structures. Every class of Markov equivalent CGs (that is CGs describing ..."
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Cited by 11 (3 self)
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The class of chain graphs (CGs) involving both undirected graphs (= Markov networks) and directed acyclic graphs (= Bayesian networks) was introduced in middle eighties for description of probabilistic conditional independence structures. Every class of Markov equivalent CGs (that is CGs describing the same conditional independence structure) has a natural representative, which is called the largest CG. The paper presents socalled recovery algorithm, which on basis of the conditional independence structure given by a CG (in form of socalled dependency model) finds the largest CG, representing the corresponding class of Markov equivalent CGs. As a byproduct a graphical characterization of graphs, which are the largest CGs (for a class of Markov equivalent CGs) is obtained, and a simple algorithm changing every CG into the largest CG of the corresponding equivalence class is given. 1 INTRODUCTION Classic graphical approaches to description of probabilistic conditional independence stru...
Racing for Conditional Independence Inference
"... Abstract. In this article, we consider the computational aspects of deciding whether a conditional independence statement t is implied by a list of conditional independence statements L using the implication related to the method of structural imsets. We present two methods which have the interestin ..."
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Cited by 1 (0 self)
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Abstract. In this article, we consider the computational aspects of deciding whether a conditional independence statement t is implied by a list of conditional independence statements L using the implication related to the method of structural imsets. We present two methods which have the interesting complementary properties that one method performs well to prove that t is implied by L, while the other performs well to prove that t is not implied by L. However, both methods do not perform well the opposite. This gives rise to a parallel algorithm in which both methods race against each other in order to determine effectively whether t is or is not implied. Some empirical evidence is provided that suggest this racing algorithms method performs a lot better than an existing method based on socalled skeletal characterization of the respective implication. Furthermore, the method is able to handle more than five variables. 1